Method of and apparatus for independently determining the resistivity and/or dielectric constant of an earth formation

ABSTRACT

Techniques are provided to transform attenuation and phase measurements taken in conjunction with a drilling operation into independent electrical parameters such as electrical resistivity and dielectric values. The electrical parameters are correlated with background values such that resulting estimates of the electrical parameters are independent of each other. It is shown an attenuation measurement is sensitive to the resistivity in essentially the same volume of an earth formation as the corresponding phase measurement is sensitive to the dielectric constant. Further, the attenuation measurement is shown to be sensitive to the dielectric constant in essentially the same volume that the corresponding phase measurement is sensitive to the resistivity. Techniques are employed to define systems of simultaneous equations that produce more accurate measurements of the resistivity and/or the dielectric constant within the earth formation than are available from currently practiced techniques.

BACKGROUND OF THE INVENTION

[0001] 1. Field of the Invention

[0002] The present invention generally relates to a method of surveyingearth formations in a borehole and, more specifically, to a method ofand apparatus for independently determining the electrical resistivityand/or dielectric constant of earth formations during Measurement-WhileDrilling/Logging-While-Drilling and Wireline Logging operations.

[0003] 2. Description of the Related Art

[0004] Typical petroleum drilling operations employ a number oftechniques to gather information about earth formations during and inconjunction with drilling operations such as Wireline Logging,Measurement-While-Drilling (MWD) and Logging-While-Drilling (LWD)operations. Physical values such as the electrical conductivity and thedielectric constant of an earth formation can indicate either thepresence or absence of oil-bearing structures near a drill hole, or“borehole.” A wealth of other information that is useful for oil welldrilling and production is frequently derived from such measurements.Originally, a drill pipe and a drill bit were pulled from the boreholeand then instruments were inserted into the hole in order to collectinformation about down hole conditions. This technique, or “wirelinelogging,” can be expensive in terms of both money and time. In addition,wireline data may be of poor quality and difficult to interpret due todeterioration of the region near the borehole after drilling. Thesefactors lead to the development of Logging-While-Drilling (LWD). LWDoperations involve collecting the same type of information as wirelinelogging without the need to pull the drilling apparatus from theborehole. Since the data are taken while drilling, the measurements areoften more representative of virgin formation conditions because thenear-borehole region often deteriorates over time after the well isdrilled. For example, the drilling fluid often penetrates or invades therock over time, making it more difficult to determine whether the fluidsobserved within the rock are naturally occurring or drilling induced.Data acquired while drilling are often used to aid the drilling process.For example, MWD/LWD data can help a driller navigate the well so thatthe borehole is ideally positioned within an oil bearing structure. Thedistinction between LWD and MWD is not always obvious, but MWD usuallyrefers to measurements taken for the purpose of drilling the well (suchas navigation) whereas LWD is principally for the purpose of estimatingthe fluid production from the earth formation. These terms willhereafter be used synonymously and referred to collectively as“MWD/LWD.”

[0005] In wireline logging, wireline induction measurements are commonlyused to gather information used to calculate the electricalconductivity, or its inverse resistivity. See for example U.S. Pat. No.5,157,605. A dielectric wireline tool is used to determine thedielectric constant and/or resistivity of an earth formation. This istypically done using measurements which are sensitive to the volume nearthe borehole wall. See for example U.S. Pat. No. 3,944,910. In MWD/LWD,a MWD/LWD resistivity tool is typically employed. Such devices are oftencalled “propagation resistivity” or “wave resistivity” tools, and theyoperate at frequencies high enough that the measurement is sensitive tothe dielectric constant under conditions of either high resistivity or alarge dielectric constant. See for example U.S. Pat. Nos. 4,899,112and4,968,940. In MWD applications, resistivity measurements may be used forthe purpose of evaluating the position of the borehole with respect toboundaries of the reservoir such as with respect to a nearby shale bed.The same resistivity tools used for LWD may also used for MWD; but, inLWD, other formation evaluation measurements including density andporosity are typically employed.

[0006] For purposes of this disclosure, the terms “resistivity” and“conductivity” will be used interchangeably with the understanding thatthey are inverses of each other and the measurement of either can beconverted into the other by means of simple mathematical calculations.The terms “depth,” “point(s) along the borehole,” and “distance alongthe borehole axis” will also be used interchangeably. Since the boreholeaxis may be tilted with respect to the vertical, it is sometimesnecessary to distinguish between the vertical depth and distance alongthe borehole axis. Should the vertical depth be referred to, it will beexplicitly referred to as the “vertical depth.”

[0007] Typically, the electrical conductivity of an earth formation isnot measured directly. It is instead inferred from other measurementseither taken during (MWD/LWD) or after (Wireline Logging) the drillingoperation. In typical embodiments of MWD/LWD resistivity devices, thedirect measurements are the magnitude and the phase shift of atransmitted electrical signal traveling past a receiver array. See forexample U.S. Pat. Nos. 4,899,112, 4,968,940, or 5,811,973. In commonlypracticed embodiments, the transmitter emits electrical signals offrequencies typically between four hundred thousand and two millioncycles per second (0.4-2.0 MHz). Two induction coils spaced along theaxis of the drill collar having magnetic moments substantially parallelto the axis of the drill collar typically comprise the receiver array.The transmitter is typically an induction coil spaced along the axis ofa drill collar from the receiver with its magnetic moment substantiallyparallel to the axis of the drill collar. A frequently used mode ofoperation is to energize the transmitter for a long enough time toresult in the signal being essentially a continuous wave (only afraction of a second is needed at typical frequencies of operation). Themagnitude and phase of the signal at one receiving coil is recordedrelative to its value at the other receiving coil. The magnitude isoften referred to as the attenuation, and the phase is often called thephase shift. Thus, the magnitude, or attenuation, and the phase shift,or phase, are typically derived from the ratio of the voltage at onereceiver antenna relative to the voltage at another receiver antenna.

[0008] Commercially deployed MWD/LWD resistivity measurement systems usemultiple transmitters; consequently, attenuation and phase-basedresistivity values can be derived independently using each transmitteror from averages of signals from two or more transmitters. See forexample U.S. Pat. No. 5,594,343.

[0009] As demonstrated in U.S. Pat. Nos. 4,968,940 and 4,899,112, a verycommon method practiced by those skilled in the art of MWD/LWD fordetermining the resistivity from the measured data is to transform thedielectric constant into a variable that depends on the resistivity andthen to independently convert the phase shift and attenuationmeasurements to two separate resistivity values. A key assumptionimplicitly used in this practice is that each measurement senses theresistivity within the same volume that it senses the dielectricconstant. This implicit assumption is shown herein by the Applicant tobe false. This currently practiced method may provide significantlyincorrect resistivity values, even in virtually homogeneous earthformations; and the errors may be even more severe in inhomogeneousformations.

[0010] A MWD/LWD tool typically transmits a 2 MHz signal (althoughfrequencies as low as 0.4 MHz are sometimes used). This frequency rangeis high enough to create difficulties in transforming the rawattenuation and phase measurements into accurate estimates of theresistivity and/or the dielectric constant. For example, the directlymeasured values are not linearly dependent on either the resistivity orthe dielectric constant (this nonlinearity, known to those skilled inthe art as “skin-effect,” also limits the penetration of the fields intothe earth formation). In addition, it is useful to separate the effectsof the dielectric constant and the resistivity on the attenuation andphase measurements given that both the resistivity and the dielectricconstant typically vary spatially within the earth formation. If theeffects of both of these variables on the measurements are notseparated, the estimate of the resistivity can be corrupted by thedielectric constant, and the estimate of the dielectric constant can becorrupted by the resistivity. Essentially, the utility of separating theeffects is to obtain estimates of one parameter that don't depend on(are independent of) the other parameter. A commonly used currentpractice relies on assuming a correlative relationship between theresistivity and dielectric constant (i.e., to transform the dielectricconstant into a variable that depends on the resistivity) and thencalculating resistivity values independently from the attenuation andphase shift measurements that are consistent the correlativerelationship. Differences between the resistivity values derived fromcorresponding phase and attenuation measurements are then ascribed tospatial variations (inhomogeneities) in the resistivity over thesensitive volume of the phase shift and attenuation measurements. Seefor example U.S. Pat. Nos. 4,899,112 and 4,968,940. An implicit andinstrumental assumption in this method is that the attenuationmeasurement senses both the resistivity and dielectric constant withinthe same volume, and that the phase shift measurement senses bothvariables within the same volume (but not the same volume as theattenuation measurement). See for example U.S. Pat. Nos. 4,899,112 and,4,968,940. These assumptions facilitate the independent determination ofa resistivity value from a phase measurement and another resistivityvalue from an attenuation measurement. However, as is shown later, theimplicit assumption mentioned above is not true; so, the resultsdetermined using such algorithms are questionable. Methods are hereindisclosed to determine two resistivity values from a phase and anattenuation measurement do not use the false assumptions of the abovementioned prior art.

[0011] Another method for determining the resistivity and/or dielectricconstant is to assume a model for the measurement apparatus in, forexample, a homogeneous medium (no spatial variation in either theresistivity or dielectric constant) and then to determine values for theresistivity and dielectric constant that cause the model to agree withthe measured phase shift and attenuation data. The resistivity anddielectric constant determined by the model are then correlated to theactual parameters of the earth formation. This method is thought to bevalid only in a homogeneous medium because of the implicit assumptionmentioned in the above paragraph. A recent publication by P. T. Wu, J.R. Lovell, B. Clark, S. D. Bonner, and J. R. Tabanou entitled“Dielectric-Independent 2-MHz Propagation Resistivities” (SPE 56448,1999) (hereafter referred to as “Wu”) demonstrates that such assumptionsare used by those skilled in the art. For example, Wu states: “Onefundamental assumption in the computation of Rex is an uninvadedhomogeneous formation. This is because the phase shift and attenuationinvestigate slightly different volumes.” It is shown herein by Applicantthat abandoning the false assumptions applied in this practice resultsin estimates of one parameter (i.e., the resistivity or dielectricconstant) that have no net sensitivity to the other parameter. Thisdesirable and previously unknown property of the results is very usefulbecause earth formations are commonly inhomogeneous.

[0012] Wireline dielectric measurement tools commonly use electricalsignals having frequencies in the range 20 MHz-1.1 GHz. In this range,the skin-effect is even more severe, and it is even more useful toseparate the effects of the dielectric constant and resistivity. Thoseskilled in the art of dielectric measurements have also falsely assumedthat a measurement (either attenuation or phase) senses both theresistivity and dielectric constant within the same volume. The designof the measurement equipment and interpretation of the data both reflectthis. See for example U.S. Pat. Nos. 4,185,238 and 4,209,747.

[0013] Wireline induction measurements are typically not attenuation andphase, but instead the real (R) and imaginary (X) parts of the voltageacross a receiver antenna which consists of several induction coils inelectrical series. For the purpose of this disclosure, the R-signal fora wireline induction measurement corresponds to the phase measurement ofa MWD/LWD resistivity or wireline dielectric tool, and the X-signal fora wireline induction measurement corresponds to the attenuationmeasurement of a MWD/LWD resistivity or wireline dielectric device.Wireline induction tools typically operate using electrical signals atfrequencies from 8-200 kHz (most commonly at approximately 20 kHz). Thisfrequency range is too low for significant dielectric sensitivity innormally encountered cases; however, the skin-effect can corrupt thewireline induction measurements. As mentioned above, the skin-effectshows up as a non-linearity in the measurement as a function of theformation conductivity, and also as a dependence of the measurementsensitivity values on the formation conductivity. Estimates of theformation conductivity from wireline induction devices are often derivedfrom data processing algorithms which assume the tool response functionis the same at all depths within the processing window. The techniquesof this disclosure can be applied to wireline induction measurements forthe purpose of deriving resistivity values without assuming the toolresponse function is the same at all depths within the processing windowas is done in U.S. Pat. No. 5,157,605. In order to make such anassumption, a background conductivity, σ, that applies for the datawithin the processing window is commonly used. Practicing a disclosedembodiment reduces the dependence of the results on the accuracy of theestimates for the background parameters because the backgroundparameters are not required to be the same at all depths within theprocessing window. In addition, practicing appropriate embodiments ofApplicant's techniques discussed herein reduces the need to performsteps to correct wireline induction data for the skin effect.

SUMMARY OF THE INVENTION

[0014] Techniques are provided to transform attenuation and phasemeasurements taken in conjunction with a drilling operation intoquantities suitable for producing more accurate electrical conductivityand/or dielectric constant values. The electrical conductivity anddielectric constant values are interpreted to provide information suchas the presence or absence of hydrocarbons within an earth formationpenetrated by the drilling operation. The techniques can be applied toWireline Logging, Logging-While-Drilling (LWD) andMeasurement-While-Drilling (MWD) operations.

[0015] As explained above, current data processing practices in thefield of MWD/LWD and wireline dielectric logging are based upon theassumption that an attenuation measurement is sensitive to theresistivity value of an earth formation in the same volume as theattenuation is sensitive to the dielectric constant. Current dataprocessing practices are also based upon the assumption that the phasemeasurement is sensitive to the resistivity in the same volume of theearth formation as it is sensitive to the dielectric constant, but thatthis volume of the phase measurement is different from that of theattenuation measurement. These assumptions, referred to herein as the“old assumptions,” are shown to be false. In fact, the attenuationsenses the resistivity and the dielectric constant in different volumes;and the phase shift senses the resistivity and the dielectric constantin different volumes. However, the attenuation measurement is shown tobe sensitive to the resistivity in essentially the same volume as thephase measurement is sensitive to the dielectric constant. Further, theattenuation measurement is shown to be sensitive to the dielectricconstant in essentially the same volume that the phase measurement issensitive to the resistivity.

[0016] By employing these new-found relationships among the attenuation,phase, resistivity and dielectric constant, systems of simultaneousequations are provided that produce more accurate measurements of theresistivity and/or the dielectric constant within an earth formationthan measurements produced using the old assumptions. In someembodiments, the equations are manipulated in a manner that provides aresistivity component that is relatively insensitive to the dielectricconstant and provides a dielectric constant component that is relativelyinsensitive to the resistivity value. Thus, more accurate and/or robustcalculations of both the resistivity and the dielectric constant areproduced.

[0017] The disclosed techniques are also applied to more complicatedscenarios wherein multiple transmitters (possibly driven at multiplefrequencies), multiple receivers, data acquired at multiple depths, orcombinations of the above are considered simultaneously. Solving aprescribed system of equations using this disclosed embodiment resultsin estimates of an average conductivity value and an average dielectricconstant value within a volume of the earth formation corresponding tointegrated averages of each parameter over said volume. In general, theresistivity and dielectric constant are expanded using basis functionsto characterize the spatial dependence of these variables. A system ofequations which can be solved for the coefficients of this expansion isgiven. Once the coefficients are determined, the spatial dependence ofboth the resistivity and dielectric constant are known.

[0018] One disclosed embodiment employs a transformation to convert thedielectric constant into a variable that depends on the resistivitythereby eliminating the dielectric constant as a variable. Tworesistivity estimates from a phase shift and an attenuation measurementare then calculated. These estimates are not determined independently asis done in the prior art because the equations solved to obtain theestimates are coupled. The manner in which these equations are coupledis consistent with the actual sensitivities of each measurement (i.e.,the phase shift and attenuation) with respect to changes in eachvariable (i.e., the resistivity and dielectric constant). Unlikeprevious MWD/LWD processing techniques, some disclosed embodimentsaccount for dielectric effects, provide for inhomogeneities, and treateach signal as a complex-valued function of the conductivity anddielectric constant, assuring that estimates of each variable are notcorrupted by effects of the other. Treating both the measurements(attenuation and phase) and the variables (conductivity and dielectricconstant) mathematically as complex-valued functions is a useful featureof the disclosed embodiments. Good results from the disclosedembodiments are produced by using the new found relationship regardingthe volume of investigation of each measurement with respect to theconductivity and the dielectric constant. In contrast, the oldassumptions imply that these results are impracticable. This is readilyevident from discussions in U.S. Pat. Nos. 4,185,238; 4,209,747;4,899,112; and 4,968,940.

BRIEF DESCRIPTION OF THE DRAWINGS

[0019] A better understanding of the present invention can be obtainedwhen the following detailed description of some preferred embodiments isconsidered in conjunction with the following drawings, in which:

[0020]FIG. 1 is a plot of multiple laboratory measurements on rocksamples representing the relationship between the conductivity and thedielectric constant in a variety of geological media;

[0021]FIG. 2 illustrates the derivation of a sensitivity function inrelation to an exemplary one-transmitter, one-receiver MWD/LWDresistivity tool;

[0022]FIG. 3 illustrates an exemplary one-transmitter, two-receiverMWD/LWD tool commonly referred to as an uncompensated measurementdevice;

[0023]FIG. 4 illustrates an exemplary two-transmitter, two-receiverMWD/LWD tool, commonly referred to as a compensated measurement device;

[0024]FIGS. 5a, 5 b, 5 c and 5 d are exemplary sensitivity functionplots for Deep and Medium attenuation and phase shift measurements;

[0025]FIGS. 6a, 6 b, 6 c and 6 d are plots of the sensitivity functionsfor the Deep and Medium measurements of FIGS. 5a, 5 b, 5 c and 5 drespectively transformed according to the techniques of a disclosedembodiment;

[0026]FIG. 7 is a portion of a table of background medium values andintegral values employed in a disclosed embodiment;

[0027]FIG. 8 is a plot of attenuation and phase as a function ofresistivity and dielectric constant; and

[0028]FIG. 9 is a flowchart of a process, that implements the techniquesof a disclosed embodiment.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

[0029] Some of the disclosed embodiments are relevant to both wirelineinduction and dielectric applications, as well asMeasurement-While-Drilling and Logging-While-Drilling (MWD/LWD)applications. Turning now to the figures, FIG. 1 is a plot ofmeasurements of the conductivity and dielectric constant determined bylaboratory measurements on a variety of rock samples from differentgeological environments. The points 121 through 129 represent measuredvalues of conductivity and dielectric constant (electrical parameters)for carbonate and sandstone earth formations. For instance, the point126 represents a sample with a conductivity value of 0.01 (10⁻²) siemensper meter (S/m) and a relative dielectric constant of approximately 22.It should be noted that both the conductivity scale and the ,dielectricscale are logarithmic scales; so, the data would appear to be much morescattered if they were plotted on linear scales.

[0030] The maximum boundary 111 indicates the maximum dielectricconstant expected to be observed at each corresponding conductivity. Ina similar fashion, the minimum boundary 115 represents the minimumdielectric constant expected to be observed at each correspondingconductivity. The points 122 through 128 represent measured values thatfall somewhere in between the minimum boundary 115 and the maximumboundary 111. A median line 113 is a line drawn so that half the points,or points 121 through 124 are below the median line 113 and half thepoints, or points 126 through 129 are above the median line 113. Thepoint 125 falls right on top of the median line 113.

[0031] An elemental measurement between a single transmitting 205 and asingle receiving coil 207 is difficult to achieve in practice, but it isuseful for describing the sensitivity of the measurement to variationsof the conductivity and dielectric constant within a localized volume225 of an earth formation 215. FIG. 2 illustrates in more detailspecifically what is meant by the term “sensitivity function,” alsoreferred to as a “response function” or “geometrical factors.”Practitioners skilled in the art of wireline logging,Measurement-While-Drilling (MWD) and Logging-While-Drilling (LWD) arefamiliar with how to generalize the concept of a sensitivity function toapply to realistic measurements from devices using multiple transmittingand receiving antennas. Typically a MWD/LWD resistivity measurementdevice transmits a signal using a transmitter coil and measures thephase and magnitude of the signal at one receiver antenna 307 relativeto the values of the phase and the magnitude at another receiver antenna309 within a borehole 301 (FIG. 3)., These relative values are commonlyreferred to as the phase shift and attenuation. It should be understoodthat one way to represent a complex signal with multiple components isas a phasor signal.

[0032] Sensitivity Functions

[0033]FIG. 2 illustrates an exemplary single transmitter, singlereceiver MWD/LWD resistivity tool 220 for investigating an earthformation 215. A metal shaft, or “mandrel,” 203 is incorporated withinthe drill string (the drill string is not shown, but it is a series ofpipes screwed together with a drill bit on the end), inserted into theborehole 201, and employed to take measurements of an electrical signalthat originates at a transmitter 205 and is sensed at a receiver 207.The measurement tool is usually not removed from the well until thedrill string is removed for the purpose of changing drill bits orbecause drilling is completed. Selected data from the tool aretelemetered to the surface while drilling. All data are typicallyrecorded in memory banks for retrieval after the tool is removed fromthe borehole 201. Devices with a single transmitter and a singlereceiver are usually not used in practice, but they are helpful fordeveloping concepts such as that of the sensitivity function. Schematicdrawings of simple, practical apparatuses are shown in FIGS. 3 and 4.

[0034] In a wireline operation, the measurement apparatus is connectedto a cable (known as a wireline), lowered into the borehole 201, anddata are acquired. This is done typically after the drilling operationis finished. Wireline induction tools measure the real (R) and imaginary(X) components of the receiver 207 signal. The R and X-signalscorrespond to the phase shift and attenuation measurements respectively.In order to correlate the sensitivity of the phase shift and attenuationmeasurements to variations in the conductivity and dielectric constantof the earth formation 215 at different positions within the earthformation, the conductivity and dielectric constant within a smallvolume P 225 are varied. For simplicity, the volume P 225 is a solid ofrevolution about the tool axis (such a volume is called atwo-dimensional volume). The amount the phase and attenuationmeasurements change relative to the amount the conductivity anddielectric constant changed within P 225 is essentially the sensitivity.The sensitivity function primarily depends on the location of the pointP 225 relative to the locations of the transmitter 205 and receiver 207,on the properties of the earth formation 215, and on the excitationfrequency. It also depends on other variables such as the diameter andcomposition of the mandrel 203, especially when P is near the surface ofthe mandrel 203.

[0035] Although the analysis is carried out in two-dimensions, theimportant conclusions regarding the sensitive volumes of phase shift andattenuation measurements with respect to the conductivity and dielectricconstant hold in three-dimensions. Consequently, the scope of thisapplication is not limited to two-dimensional cases. This is discussedmore in a subsequent section entitled, “ITERATIVE FORWARD MODELING ANDDIPPING BEDS.”

[0036] The sensitivity function can be represented as a complex numberhaving a real and an imaginary part. In the notation used below, S,denotes a complex sensitivity function, and its real part is S′, and itsimaginary part is S″. Thus, S=S′+iS″, in which the imaginary numberi={square root}{square root over (−1)}. The quantities S′ and S″ arecommonly referred to as geometrical factors or response functions. Thevolume P 225 is located a distance p in the radial direction from thetool's axis and a distance z in the axial direction from the receiver206. S′ represents the sensitivity of attenuation to resistivity and thesensitivity of phase shift to dielectric constant. Likewise, S″represents the sensitivity of attenuation to dielectric constant and thesensitivity of phase shift to resistivity. The width of the volume P 225is Δρ 211 and the height of the volume P 225 is Δz 213. The quantity S′,or the sensitivity of attenuation to resistivity, is calculated bydetermining the effect a change in the conductivity (reciprocal ofresistivity) in volume P 225 from a prescribed background value has onthe attenuation of a signal between the transmitter 205 and the receiver207, assuming the background conductivity value is otherwise unperturbedwithin the entire earth formation 215. In a similar fashion, S″, or thesensitivity of the phase to the resistivity, is calculated bydetermining the effect a change in the conductivity value in the volumeP 225 from an assumed background conductivity value has on the phase ofthe signal between the transmitter 205 and the receiver 207, assumingthe background parameters are otherwise unperturbed within the earthformation 215. Alternatively, one could determine S′ and S″ bydetermining the effect a change the dielectric constant within thevolume P 225 has on the phase and attenuation, respectively. When thesensitivities are determined by considering a perturbation to thedielectric constant value within the volume P 225, it is apparent thatthe sensitivity of the attenuation to changes in the dielectric constantis the same as the sensitivity of the phase to the conductivity. It isalso apparent that the sensitivity of the phase to the dielectricconstant is the same as the sensitivity of the attenuation to theconductivity. By simultaneously considering the sensitivities of boththe phase and attenuation measurement to the dielectric constant and tothe conductivity, the Applicant shows a previously unknown relationshipbetween the attenuation and phase shift measurements and theconductivity and dielectric constant values. By employing thispreviously unknown relationship, the Applicant provides techniques thatproduce better estimates of both the conductivity and the dielectricconstant values than was previously available from those with skill inthe art. The sensitivity functions S′ and S″ and their relation to thesubject matter of the Applicant's disclosure is explained in more detailbelow in conjunction with FIGS. 5a-d and FIGS. 6a-d.

[0037] In the above, sensitivities to the dielectric constant werereferred to. Strictly speaking, the sensitivity to the radian frequencyω times the dielectric constant should have been referred to. Thisdistinction is trivial to those skilled in the art.

[0038] In FIG. 2, if the background conductivity (reciprocal ofresistivity) of the earth formation 215 is σ₀ and the backgrounddielectric constant of the earth formation 215 is ε₀, then the ratio ofthe receiver 207 voltage to the transmitter 205 current in thebackground medium can be expressed as Z_(RT) ⁰, where R stands for thereceiver 207 and T stands for the transmitter 205. Hereafter, a numberedsubscript or superscript such as the ‘0’ is merely used to identify aspecific incidence of the corresponding variable or function. If anexponent is used, the variable or function being raised to the powerindicated by the exponent will be surrounded by parentheses and theexponent will be placed outside the parentheses. For example (L₁)³ wouldrepresent the variable L₁ raised to the third power.

[0039] When the background conductivity σ₀ and/or dielectric constant ε₀are replaced new values σ₁ and/or ε₁ in the volume P 225, the ratiobetween the receiver 207 voltage to the transmitter 205 current isrepresented by Z_(RT) ¹. Using the same nomenclature, a ratio between avoltage at a hypothetical receiver placed in the volume P 225 and thecurrent at the transmitter 205 can be expressed as Z_(PT) ⁰. Inaddition, a ratio between the voltage at the receiver 207 and a currentat a hypothetical transmitter in the volume P 225 can be expressed asZ_(RP) ⁰. Using the Born approximation, it can be shown that,$\frac{Z_{RT}^{1}}{Z_{RT}^{0}} = {1 + {{S\left( {T,R,P} \right)}\Delta \overset{\sim}{\sigma}{\Delta\rho\Delta}\quad z}}$

[0040] where the sensitivity function, defined as S(T,R,P), is${S\left( {T,R,P} \right)} = {- \frac{Z_{RP}^{0}Z_{PT}^{0}}{2{\pi\rho}\quad Z_{RT}^{0}}}$

[0041] in which Δ{tilde over (σ)}={tilde over (σ)}₁−{tilde over(σ)}₀=(σ₁−σ₀)+iω(ε₁−ε₀), and the radian frequency of the transmittercurrent is ω=2πƒ. A measurement of this type, in which there is just onetransmitter 205 and one receiver 207, is defined as an “elemental”measurement. It should be noted that the above result is also valid ifthe background medium parameters vary spatially within the earthformation 215. In the above equations, both the sensitivity functionS(T, R, P) and the perturbation As are complex-valued. Some disclosedembodiments consistently treat the measurements, their sensitivities,and the parameters to be estimated as complex-valued functions. This isnot done in the prior art.

[0042] The above sensitivity function of the form S(T, R, P) is referredto as a 2-D (or two-dimensional) sensitivity function because the volumeΔρΔz surrounding the point P 225, is a solid of revolution about theaxis of the tool 201. Because the Born approximation was used, thesensitivity function S depends only on the properties of the backgroundmedium because it is assumed that the same field is incident on thepoint P(ρ, z) even though the background parameters have been replacedby {tilde over (σ)}₁.

[0043]FIG. 3 illustrates an exemplary one-transmitter, two-receiverMWD/LWD resistivity measurement apparatus 320 for investigating an earthformation 315. Due to its configuration, the tool 320 is defined as an“uncompensated” device and collects uncompensated measurements from theearth formation 315. For the sake of simplicity, a borehole is notshown. This measurement tool 320 includes a transmitter 305 and tworeceivers 307 and 309, each of which is incorporated into a metalmandrel 303. Typically, the measurement made by such a device is theratio of the voltages at receivers 307 and 309. In this example, usingthe notation described above in conjunction with FIG. 2, the sensitivityfunction S(T, R, R′, P) for the uncompensated device can be shown to bethe difference between the elemental sensitivity functions S(T,R,P) andS(T, R′, P), where T represents the transmitter 305, R represents thereceiver 307, R′ represents the receiver 309, and P represents a volume(not shown) similar to the volume P 225 of FIG. 2.

[0044] For wireline induction measurements, the voltage at the receiverR is subtracted from the voltage at the receiver R′, and the positionand number of turns of wire for R are commonly chosen so that thedifference, in the voltages at the two receiver antennas is zero whenthe tool is in a nonconductive medium., For MWD/LWD resistivity andwireline dielectric constant measurements, the voltage at the receiverR, or V_(R), and the voltage at the receiver R′, or V_(R), are examinedas the ratio V_(R)/V_(R′). In either case, it can be shown that

S(T,R,R′,P)=S(T,R,P)−S(T,R′,P).

[0045] The sensitivity for an uncompensated measurement is thedifference between the sensitivities of two elemental measurements suchas S(T,R,P) and S(T,R′,P) calculated as described above in conjunctionwith FIG. 2.

[0046]FIG. 4 illustrates an exemplary two-transmitter, two-receiverMWD/LWD resistivity tool 420. Due to its configuration (transmittersbeing disposed symmetrically), the tool 420 is defined as a“compensated” tool and collects compensated measurements from an earthformation 415. The tool 420 includes two transmitters 405 and 411 andtwo receivers 407 and 409, each of which is incorporated into a metalmandrel or collar 403. Each compensated measurement is the geometricmean of two corresponding uncompensated measurements. In other words,during a particular timeframe, the tool 420 performs two uncompensatedmeasurements, one employing transmitter 405 and the receivers 407 and409 and the other employing the transmitter 411 and the receivers 409and 407. These two uncompensated measurements are similar to theuncompensated measurement described above in conjunction with FIG. 3.The sensitivity function S of the tool 420 is then defined as thearithmetic average of the sensitivity functions for each of theuncompensated measurements. Another way to describe this relationship iswith the following formula:${S\left( {T,R,R^{\prime},T^{\prime},P} \right)} = {\frac{1}{2}\left\lbrack {{S\left( {T,R,R^{\prime},P} \right)} + {S\left( {T^{\prime},R^{\prime},R,P} \right)}} \right\rbrack}$

[0047] where T represents transmitter 405, T′ represents transmitter411, R represents receiver 407, R′ represents receiver 409 and Prepresents a small volume of the earth formation similar to 225 (FIG.2).

[0048] The techniques of the disclosed embodiments are explained interms of a compensated tool such as the tool 420 and compensatedmeasurements such as those described in conjunction with FIG. 4.However, it should be understood that the techniques also apply touncompensated tools such as the tool 320 and uncompensated measurementsdescribed above in conjunction with FIG. 3 and elemental tools such asthe tool 220 and elemental measurements such as those described above inconjunction with FIG. 2. In addition, the techniques are applicable foruse in a wireline system, a system that may not incorporate itstransmitters and receivers into a metal mandrel, but may rather affix atransmitter and a receiver to a tool made of a non-conducting materialsuch as fiberglass. The wireline induction frequency is typically toolow for dielectric effects to be significant. Also typical for wirelineinduction systems is to select the position and number of turns ofgroups of receiver antennas so that there is a null signal in anonconductive medium. When this is done, Z_(RT) ⁰=0 if {tilde over(σ)}₀=0. As a result, it is necessary to multiply the sensitivity andother quantities by Z_(RT) ⁰ to use the formulation given here in suchcases.

[0049] The quantity Z_(RT) ¹/Z_(RT) ⁰ can be expressed as a complexnumber which has both a magnitude and a phase (or alternatively real andimaginary parts). To a good approximation, the raw attenuation value(which corresponds to the magnitude) is:${{\frac{Z_{RT}^{1}}{Z_{RT}^{0}}} \approx {{Re}{\frac{Z_{RT}^{1}}{Z_{RT}^{0}}}}} = {{1 + {{{Re}\left\lbrack {{S\left( {T,R,P} \right)}\Delta \overset{\sim}{\sigma}} \right\rbrack}{\Delta\rho\Delta}\quad z}} = {1 + {\left\lbrack {{S^{\prime}{\Delta\sigma}} - {S^{''}{\omega\Delta ɛ}}} \right\rbrack {\Delta\rho\Delta}\quad z}}}$

[0050] where the function Re[•] denotes the real part of its argument.Also, to a good approximation, the raw phase shift value is:${{{phase}\left( \frac{Z_{RT}^{1}}{Z_{RT}^{0}} \right)} \approx {{Im}\left\lbrack \frac{Z_{RT}^{1}}{Z_{RT}^{0}} \right\rbrack}} = {{{{Im}\left\lbrack {{S\left( {T,R,P} \right)}\Delta \overset{\sim}{\sigma}} \right\rbrack}{\Delta\rho\Delta}\quad z} = {\left\lbrack {{S^{''}{\Delta\sigma}} + {S^{\prime}{\omega\Delta ɛ}}} \right\rbrack {\Delta\rho\Delta}\quad z}}$

[0051] in which Im[•] denotes the imaginary part of its argument, S(T,R, P)=S′+iS″, Δσ=σ₁−σ₀, and Δε=ε₁−ε₀. For the attenuation measurement,S′ is the sensitivity to the resistivity and S″ is the sensitivity tothe dielectric constant. For the phase shift measurement, S′ is thesensitivity to the dielectric constant and S″ is the sensitivity to theresistivity. This is apparent because S′ is the coefficient of Δσ in theequation for attenuation, and it is also the coefficient of ωΔε in theequation for the phase shift. Similarly, S″ is the coefficient of Δσ inthe equation for the phase shift, and it is also the coefficient for−ωΔε in the equation for attenuation. This implies that the attenuationmeasurement senses the resistivity in the same volume as the phase shiftmeasurement senses the dielectric constant and that the phase shiftmeasurement senses the resistivity in the same volume as the attenuationmeasurement senses the dielectric constant. In the above, we havereferred to sensitivities to the dielectric constant. Strictly speaking,the sensitivity to the radian frequency ω times the dielectric constantAc should have been referred to. This distinction is trivial to thoseskilled in the art.

[0052] The above conclusion regarding the volumes in which phase andattenuation measurements sense the resistivity and dielectric constantfrom Applicant's derived equations also follows from a well known resultfrom complex variable theory known in that art as the Cauchy-Reimannequations. These equations provide the relationship between thederivatives of the real and imaginary parts of an analytic complexfunction with respect to the real and imaginary parts of the function'sargument.

[0053]FIGS. 5a, 5 b, 5 c and 5 d can best be described and understoodtogether. In all cases, the mandrel diameter is 6.75 inches, thetransmitter frequency is 2 MHz, and the background medium ischaracterized by a conductivity of σ₀=0.01S/m and a relative dielectricconstant of ε₀=10. The data in FIGS. 5a and 5 c labeled “MediumMeasurement” are for a compensated type of design shown in FIG. 4. Theexemplary distances between transmitter 405 and receivers 407 and 409are 20 and 30 inches, respectively. Since the tool is symmetric, thedistances between transmitter 411 and receivers 409 and 407 are 20 and30 inches, respectively. The data in FIGS. 5b and 5 d labeled “DeepMeasurement” are also for a compensated tool as shown in FIG. 4, butwith exemplary transmitter-receiver spacings of 50 and 60 inches. Eachplot shows the sensitivity of a given measurement as a function ofposition within the formation. The term sensitive volume refers to theshape of each plot as well as its value at any point in the formation.The axes labeled “Axial Distance” refer to the coordinate along the axisof the tool with zero being the geometric mid-point of the antenna array(halfway between receivers 407 and 409) to a given point in theformation. The axes labeled “Radial Distance” refer to the radialdistance from the axis of the tool to a given point in the formation.The value on the vertical axis is actually the sensitivity value for theindicated measurement. Thus, FIG. 5a is a plot of a sensitivity functionthat illustrates the sensitivity of the “Medium” phase shift measurementin relation to changes in the resistivity as a function of the locationof the point P 225 in the earth formation 215 (FIG. 2). If themeasurement of phase shift changes significantly in response to changingthe resistivity from its background value, then phase shift isconsidered relatively sensitive to the resistivity at the point P 225.If the measurement of phase shift does not change significantly inresponse to changing the resistivity, then the phase shift is consideredrelatively insensitive at the point P 225. Based upon the relationshipdisclosed herein, FIG. 5a also illustrates the sensitivity of the“Medium” attenuation measurement in relation to changes in dielectricconstant values. Note that the dimensions of the sensitivity on thevertical axes is ohms per meter (Ω/m) and distances on the horizontalaxes are listed in inches. In a similar fashion, FIG. 5b is a plot ofthe sensitivity of the attenuation measurement to the resistivity. Basedon the relationship disclosed herein, FIG. 5b is also the sensitivity ofa phase shift measurement to a change in the dielectric constant. FIGS.5b and 5 d have the same descriptions as FIGS. 5a and 5 c, respectively,but FIGS. 5b and 5 d are for the “Deep Measurement” with the antennaspacings described above.

[0054] Note that the shape of FIG. 5a is very dissimilar to the shape ofFIG. 5c. This means that the underlying measurements are sensitive tothe variables in different volumes. For example, the Medium phase shiftmeasurement has a sensitive volume characterized by FIG. 5a for theresistivity, but this measurement has the sensitive volume shown in FIG.5c for the dielectric constant. As discussed below, it is possible totransform an attenuation and a phase shift measurement to a complexnumber which has the following desirable properties: 1) its real part issensitive to the resistivity in the same volume that the imaginary partis sensitive to the dielectric constant; 2) the real part has no netsensitivity to the dielectric constant; and, 3) the imaginary part hasno net sensitivity to the resistivity. In addition, the transformationis generalized to accommodate multiple measurements acquired at multipledepths. The generalized method can be used to produce independentestimates of the resistivity and dielectric constant within a pluralityof volumes within the earth formation.

[0055] Transformed Sensitivity Functions and Transformation of theMeasurements

[0056] For simplicity, the phase shift and attenuation will not be used.Hereafter, the real and imaginary parts of measurement will be referredto instead. Thus,

w=w′+iw″

w′=(10)^(db/20)×cos(θ)

w″=(10)^(dB/20)×sin(θ)

[0057] where w′ is the real part of w, w″ is the imaginary part of w, iis the square root of the integer −1, dB is the attenuation in decibels,and e is the phase shift in radians.

[0058] The equations that follow can be related to the sensitivityfunctions described above in conjunction with FIG. 2 by definingvariables w₁=Z_(RT) ¹ and w₀=Z_(RT) ⁰. The variable w₁ denotes an actualtool measurement in the earth formation 215. The variable w₀ denotes theexpected value for the tool measurement in the background earthformation 215. For realistic measurement devices such as those describedin FIGS. 3 and 4, the values for w₁ and w₀ would be the voltage ratiosdefined in the detailed description of FIGS. 3 and 4. In one embodiment,the parameters for the background medium are determined and then used tocalculate value of w₀ using a mathematical model to evaluate the toolresponse in the background medium. One of many alternative methods todetermine the background medium parameters is to estimate w₀ directlyfrom the measurements, and then to determine the background parametersby correlating w₀ to a model of the tool in the formation which has thebackground parameters as inputs.

[0059] As explained in conjunction with FIG. 2, the sensitivity functionS relates the change in the measurement to a change in the mediumparameters such as resistivity and dielectric constant within a smallvolume 225 of the earth formation 215 at a prescribed location in theearth formation 225, or background medium. A change in measurements dueto small variations in the medium parameters at a range of locations canbe calculated by integrating the responses from each such volume in theearth formation 215. Thus, if Δ{tilde over (σ)} is defined for a largenumber of points ρ,z, then

Z _(RT) ¹ =Z _(RT) ⁰(1+I[SΔ{tilde over (σ)}])

[0060] in which I is a spatial integral function further defined asI[F] = ∫_(−∞)^(+∞)  z∫₀^(+∞)  ρ  F(ρ, z)

[0061] where F is a complex function.

[0062] Although the perturbation from the background medium, Δ{tildeover (σ)} is a function of position, parameters of a hypothetical,equivalent homogeneous perturbation (meaning that no spatial variationsare assumed in the difference between the resistivity and dielectricconstant and values for both of these parameters in the backgroundmedium) can be determined by assuming the perturbation is not a functionof position and then solving for it. Thus,

Δ{circumflex over (σ)}I[S]=I[SΔ{tilde over (σ)}]

[0063] where Δ{circumflex over (σ)} represents the parameters of theequivalent homogeneous perturbation. From the previous equations, it isclear that${\Delta \hat{\sigma}} = {\frac{I\left\lbrack {S\quad \Delta \overset{\sim}{\sigma}} \right\rbrack}{I\lbrack S\rbrack} = {{I\left\lbrack {\hat{S}\Delta \overset{\sim}{\sigma}} \right\rbrack} = {\frac{1}{I\lbrack S\rbrack}\left( {\frac{w_{1}}{w_{0}} - 1} \right)\quad {and}}}}$$\hat{S} = \frac{S}{I\lbrack S\rbrack}$

[0064] where Δ{circumflex over (σ)} is the transformed measurement (itis understood that Δ{circumflex over (σ)} is also the equivalenthomogeneous perturbation and that the terms transformed measurement andequivalent homogeneous perturbation will be used synonymously), Ŝ is thesensitivity function for the transformed measurement, and Ŝ will bereferred to as the transformed sensitivity function. In the above, w₁ isthe actual measurement, and w₀ is the value assumed by the measurementin the background medium. An analysis of the transformed sensitivityfunction Ŝ, shows that the transformed measurements have the followingproperties: 1) the real part of Δ{circumflex over (σ)} is sensitive tothe resistivity in the same volume that its imaginary part is sensitiveto the dielectric constant; 2) the real part of Δ{circumflex over (σ)}has no net sensitivity to the dielectric constant; and, 3) the imaginarypart of Δ{circumflex over (σ)} has no net sensitivity to theresistivity. Details of this analysis will be given in the next fewparagraphs.

[0065] The techniques of the disclosed embodiment can be further refinedby introducing a calibration factor c (which is generally a complexnumber that may depend on the temperature of the measurement apparatusand other environmental variables) to adjust for anomalies in thephysical measurement apparatus. In addition, the term, w_(bh) can beintroduced to adjust for effects caused by the borehole 201 on themeasurement. With these modifications, the transformation equationbecomes${\Delta \hat{\sigma}} = {\frac{I\left\lbrack {S\quad \Delta \overset{\sim}{\sigma}} \right\rbrack}{I\lbrack S\rbrack} = {{I\left\lbrack {\hat{S}\Delta \overset{\sim}{\sigma}} \right\rbrack} = {\frac{1}{I\lbrack S\rbrack}{\left( {\frac{{cw}_{1} - w_{bh}}{w_{0}} - 1} \right).}}}}$

[0066] The sensitivity function for the transformed measurement isdetermined by applying the transformation to the original sensitivityfunction, S. Thus,$\hat{S} = {{{\hat{S}}^{\prime} + {i{\hat{S}}^{''}}} = {\frac{S}{I\lbrack S\rbrack} = {\frac{{S^{\prime}{I\left\lbrack S^{\prime} \right\rbrack}} + {S^{''}{I\left\lbrack S^{''} \right\rbrack}}}{{{I\lbrack S\rbrack}}^{2}} + {{\frac{{S^{''}{I\left\lbrack S^{\prime} \right\rbrack}} - {S^{\prime}{I\left\lbrack S^{''} \right\rbrack}}}{{{I\lbrack S\rbrack}}^{2}}.}}}}}$

[0067] Note that I[Ŝ]=I[Ŝ′]=1 because I[Ŝ″]=0. The parameters for theequivalent homogeneous perturbation are

Δ{circumflex over (σ)}′={circumflex over (σ)}₁−σ₀ =I[Ŝ′Δσ]−I[Ŝ″ωΔε]

Δ{circumflex over (σ)}″=ω({circumflex over (ε)}₁−ε₀)=I[Ŝ′ωΔε]+I[Ŝ″Δσ].

[0068] The estimate for the conductivity perturbation, Δ{circumflex over(σ)}′ suppresses sensitivity (is relatively insensitive) to thedielectric constant perturbation, and the estimate of the dielectricconstant perturbation, Δ{circumflex over (σ)}″/ω suppresses sensitivityto the conductivity perturbation. This is apparent because thecoefficient of the suppressed variable is Ŝ″. In fact, the estimate forthe conductivity perturbation Δ{circumflex over (σ)}′ is independent ofthe dielectric constant perturbation provided that deviations in thedielectric constant from its background are such that I[Ŝ″ωΔε]=0. SinceI[Ŝ″]=0, this is apparently the case if ωΔε is independent of position.Likewise, the estimate for the dielectric constant perturbation given byΔ{circumflex over (σ)}″/ω is independent of the conductivityperturbation provided that deviations in the conductivity from itsbackground value are such that I[Ŝ″Δσ]=0. Since I[Ŝ″]=0, this isapparently the case if Δσ is independent of position.

[0069] Turning now to FIGS. 6a and 6 b, illustrated are plots of thesensitivity functions Ŝ′ and Ŝ″ derived from S′ and S″ for the mediumtransmitter-receiver spacing measurement shown in FIGS. 5a and 5 c usingthe transformation $\hat{S} = {\frac{S}{I\lbrack S\rbrack}.}$

[0070] The data in FIGS. 6c and 6 d were derived from the data in FIGS.5b and 5 d for the Deep T-R spacing measurement. As shown in FIGS. 6a, 6b, 6 c and 6 d, using the transformed measurements to determine theelectrical parameters of the earth formation is a substantialimprovement over the prior art. The estimates of the medium parametersare more accurate and less susceptible to errors in the estimate of thebackground medium because the calculation of the resistivity isrelatively unaffected by the dielectric constant and the calculation ofthe dielectric constant is relatively unaffected by the resistivity. Inaddition to integrating to 0, the peak values for Ŝ″ in FIGS. 6b and 6 dare significantly less than the respective peak values for Ŝ′ in FIGS.6a and 6 c. Both of these properties are very desirable because Ŝ″ isthe sensitivity function for the variable that is suppressed.

[0071] Realization of the Transformation

[0072] In order to realize the transformation, it is desirable to havevalues of I[S] readily accessible over the range of background mediumparameters that will be encountered. One way to achieve this is tocompute the values for I[S] and then store them in a lookup table foruse later. Of course, it is not necessary to store these data in such alookup table if it is practical to quickly calculate the values for I[S]on command when they are needed. In general, the values for I[S] can becomputed by directly; however, it can be shown that$\left( {{{I\lbrack S\rbrack} = {\frac{1}{w_{0}}\frac{\partial w}{\partial\overset{\sim}{\sigma}}}}} \right)_{\overset{\sim}{\sigma} = {\overset{\sim}{\sigma}}_{0}}$

[0073] where w₀ is the expected value for the measurement in thebackground medium, and the indicated derivative is calculated using thefollowing definition:$\left( {\frac{\partial w}{\partial\overset{\sim}{\sigma}}} \right)_{\overset{\sim}{\sigma} = {\overset{\sim}{\sigma}}_{0}} = {\lim\limits_{{\Delta \overset{\sim}{\sigma}}\rightarrow 0}{\cdot {\frac{{w\left( {{\overset{\sim}{\sigma}}_{0} + {\Delta \overset{\sim}{\sigma}}} \right)} - {w\left( {\overset{\sim}{\sigma}}_{0} \right)}}{\left( {{\overset{\sim}{\sigma}}_{0} + {\Delta \overset{\sim}{\sigma}}} \right) - {\overset{\sim}{\sigma}}_{0}}.}}}$

[0074] In the above formula, {tilde over (σ)}₀ may vary from point topoint in the formation 215 (the background medium may be inhomogeneous),but the perturbation Δ{tilde over (σ)} is constant at all points in theformation 215. As an example of evaluating I[S] using the above formula,consider the idealized case of a homogeneous medium with a smalltransmitter coil and two receiver coils spaced a distance L₁ and L₂ fromthe transmitter. Then,$\left( {{{w_{0} = {\left( \frac{L_{1}}{L_{2}} \right)^{3}\frac{{\exp \left( {\quad k_{0}L_{2}} \right)}\left( {1 - {\quad k_{0}L_{2}}} \right)}{{\exp \left( {\quad k_{0}L_{1}} \right)}\left( {1 - {\quad k_{0}L_{1}}} \right)}}}{{I\lbrack S\rbrack} = {\frac{1}{w_{0}}\frac{\partial w}{\partial\overset{\sim}{\sigma}}}}}} \right)_{\overset{\sim}{\sigma} = {\overset{\sim}{\sigma}}_{0}} = {\frac{\omega\mu}{2}\left( {\frac{\left( L_{2} \right)^{2}}{1 - {\quad k_{0}L_{2}}} - \frac{\left( L_{1} \right)^{2}}{1 - {\quad k_{0}L_{1}}}} \right)}$

[0075] The wave number in the background medium is k₀={squareroot}{square root over (iωμ{tilde over (σ)})}₀, the function exp(•) isthe complex exponential function where exp(1)≈2.71828, and the symbol μdenotes the magnetic permeability of the earth formation. The aboveformula for I[S] applies to both uncompensated (FIG. 3) and tocompensated (FIG. 4) measurements because the background medium hasreflection symmetry about the center of the antenna array in FIG. 4.

[0076] For the purpose of this example, the above formula is used tocompute the values for I[S]=I[S′]+iI[S″]. FIG. 7 illustrates anexemplary table 701 employed in a Create Lookup Table step 903 (FIG. 9)of the technique of the disclosed embodiment. Step 903 generates a tablesuch as table 701 including values for the integral of the sensitivityfunction over the range of variables of interest. The first two columnsof the table 701 represent the conductivity σ₀ and the dielectricconstant ε₀ of the background medium. The third and fourth columns ofthe table 701 represent calculated values for the functions I[S′] andI[S″] for a Deep measurement, in which the spacing between thetransmitter 305 receivers 307 and 309 is 50 and 60 inches, respectively.The fifth and sixth columns of the table 701 represent calculated valuesfor the functions I[S′] and I[S″] for a Medium measurement, in which thespacing between the transmitter 305 receivers 307 and 309 is 20 and 30inches, respectively. It is understood that both the frequency of thetransmitter(s) and the spacing between the transmitter(s) andreceiver(s) can be varied. Based upon this disclosure, it is readilyapparent to those skilled in the art that algorithms such as the onedescribed above can be applied to alternative measurementconfigurations. If more complicated background media are used, forexample including the mandrel with finite-diameter antennas, it may bemore practical to form a large lookup table such as table 701 but withmany more values. Instead of calculating I[S] every time a value isneeded, data would be interpolated from the table. Nonetheless, table701 clearly illustrates the nature of such a lookup table. Such a tablewould contain the values of the functions I[S′] and I[S″] for the entirerange of values of the conductivity σ₀ and the dielectric constant ε₀likely to be encountered in typical earth formations. For example, I[S′]and I[S″] could be calculated for values of ε₀ between 1 and 1000 andfor values of σ₀ between 0.0001 and 10.0. Whether calculating values forthe entire lookup table 701 or computing the I[S′] and I[S″] on commandas needed, the data is used as explained below.

[0077]FIG. 8 illustrates a chart 801 used to implement a DetermineBackground Medium Parameters step 905 (FIG. 9) of the techniques of thedisclosed embodiment. The chart 801 represents a plot of the attenuationand phase shift as a function of resistivity and dielectric constant ina homogeneous medium. Similar plots can be derived for more complicatedmedia. However, the homogeneous background media are routinely used dueto their simplicity. Well known numerical methods such as inverseinterpolation can be used to calculate an initial estimate of backgroundparameters based upon the chart 801. In one embodiment, the measuredattenuation and phase shift values are averaged over a few feet of depthwithin the borehole 201. These average values are used to determine thebackground resistivity and dielectric constant based upon the chart 801.It should be understood that background medium parameters can beestimated in a variety of ways using one or more attenuation and phasemeasurements.

[0078]FIG. 9 is a flowchart of an embodiment of the disclosedtransformation techniques that can be implemented in a software programwhich is executed by a processor of a computing system such as acomputer at the surface or a “downhole” microprocessor. Starting in aBegin Analysis step 901, control proceeds immediately to the CreateLookup Table step 903 described above in conjunction with FIG. 7. In analternative embodiment, step 903 can be bypassed and the function of thelookup table replaced by curve matching, or “forward modeling.” Controlthen proceeds to an Acquire Measured Data Step 904. Next, controlproceeds to a Determine Background Values Step 905, in which thebackground values for the background medium are determined. Step 905corresponds to the chart 801 (FIG. 8).

[0079] Control then proceeds to a Determine Integral Value step 907. TheDetermine Integral Value step 907 of the disclosed embodiment determinesan appropriate value for I[S] using the lookup table generated in thestep 903 described above or by directly calculating the I[S] value asdescribed in conjunction with FIG. 7. Compute Parameter Estimate, step909, computes an estimate for the conductivity and dielectric constantas described above using the following equation:${\Delta \hat{\sigma}} = {\frac{I\left\lbrack {S\quad \Delta \overset{\sim}{\sigma}} \right\rbrack}{I\lbrack S\rbrack} = {{I\left\lbrack {\hat{S}\Delta \overset{\sim}{\sigma}} \right\rbrack} = {\frac{1}{I\lbrack S\rbrack}{\left( {\frac{{cw}_{1} - w_{bh}}{w_{0}} - 1} \right).}}}}$

[0080] where the borehole effect and a calibration factor are taken intoaccount using the factors w_(bh) and c, respectively. The conductivityvalue plotted on the log (this is the value correlated to theconductivity of the actual earth formation) is Re(Δ{circumflex over(σ)}+{tilde over (σ)}₀) where the background medium is characterized by{tilde over (σ)}₀. The estimate for the dielectric constant can also beplotted on the log (this value is correlated to the dielectric constantof the earth formation), and this value is Im(Δ{circumflex over(σ)}+{tilde over (σ)}₀)/ω. Lastly, in the Final Depth step 911, it isdetermined whether the tool 201 is at the final depth within the earthformation 215 that will be considered in the current logging pass. Ifthe answer is “Yes,” then control proceeds to a step 921 where isprocessing is complete. If the answer in step 911 is “No,” controlproceeds to a Increment Depth step 913 where the tool 220 is moved toits next position in the borehole 201 which penetrates the earthformation 215. After incrementing the depth of the tool 220, controlproceeds to step 904 where the process of steps 904, 905, 907, 909 and911 are repeated. It should be understood by those skilled in the artthat embodiments described herein in the form a computing system or as aprogrammed electrical circuit can be realized.

[0081] Improved estimates for the conductivity and/or dielectricconstant can be determined by simultaneously considering multiplemeasurements at multiple depths. This procedure is described in moredetail below under the heading “Multiple Sensors At Multiple Depths.”

[0082] Multiple Sensors at Multiple Depths

[0083] In the embodiments described above, the simplifying assumptionthat Δ{tilde over (σ)} is not position dependent facilitates determininga value for Δ{circumflex over (σ)} associated with each measurement byconsidering only that measurement at a single depth within the well (atleast given a background value {tilde over (σ)}₀). It is possible toeliminate the assumption that Δ{tilde over (σ)} is independent ofposition by considering data at multiple depths, and in general, to alsoconsider multiple measurements at each depth. An embodiment of such atechnique for jointly transforming data from multiple MWD/LWD sensors atmultiple depths is given below. Such an embodiment can also be used forprocessing data from a wireline dielectric tool or a wireline inductiontool. Alternate embodiments can be developed based on the teachings ofthis disclosure by those skilled in the art.

[0084] In the disclosed example, the background medium is not assumed tobe the same at all depths within the processing window. In cases whereit is possible to assume the background medium is the same at all depthswithin the processing window, the system of equations to be solved is inthe form of a convolution. The solution to such systems of equations canbe expressed as a weighted sum of the measurements, and the weights canbe determined using standard numerical methods. Such means are known tothose skilled in the art, and are referred to as “deconvolution”techniques. It will be readily understood by those with skill in the artthat deconvolution techniques can be practiced in conjunction with thedisclosed embodiments without departing from the spirit of theinvention, but that the attendant assumptions are not necessary topractice the disclosed embodiments in general.

[0085] Devices operating at multiple frequencies are considered below,but multifrequency operation is not necessary to practice the disclosedembodiments. Due to frequency dispersion (i.e., frequency dependence ofthe dielectric constant and/or the conductivity value), it is notnecessarily preferable to operate using multiple frequencies. Given thedisclosed embodiments, it is actually possible to determine thedielectric constant and resistivity from single-frequency data. In fact,the disclosed embodiments can be used to determine and quantifydispersion by separately processing data sets acquired at differentfrequencies. In the below discussion, it is understood that subsets ofdata from a given measurement apparatus or even from several apparatusescan be processed independently to determine parameters of interest. Thebelow disclosed embodiment is based on using all the data availablestrictly for purpose of simplifying the discussion.

[0086] Suppose multiple transmitter-receiver spacings are used and thateach transmitter is excited using one or more frequencies. Further,suppose data are collected at multiple depths in the earth formation215. Let N denote the number of independent measurements performed ateach of several depths, where a measurement is defined as the dataacquired at a particular frequency from a particular set of transmittersand receivers as shown in FIGS. 3 or 4. Then, at each depth z_(k), avector of all the measurements can be defined as${{\overset{\_}{v}}_{k} = \left\lbrack {\left( {\frac{w_{1}}{w_{0}} - 1} \right)_{k1},\left( {\frac{w_{1}}{w_{0}} - 1} \right)_{k2},\cdots \quad,\left( {\frac{w_{1}}{w_{0}} - 1} \right)_{kN}} \right\rbrack^{T}},$

[0087] and the perturbation of the medium parameters from the backgroundmedium values associated with these measurements is

Δ{tilde over ({overscore (σ)})}=Δ{tilde over (σ)}(ρ,z)[1,1, . . .,1]^(T)

[0088] in which the superscript T denotes a matrix transpose, {overscore(v)}_(k) is a vector each element of which is a measurement, and Δ{tildeover ({overscore (σ)})} is a vector each element of which is aperturbation from the background medium associated with a correspondingelement of {overscore (v)}_(k) at the point P 225. In the above, thedependence of the perturbation, Δ{tilde over (σ)}(ρ,z)on the position ofthe point P 225 is explicitly denoted by the variables ρ and z. Ingeneral, the conductivity and dielectric constant of both the backgroundmedium and the perturbed medium depend on ρ and z; consequently, nosubscript k needs to be associated with Δ{tilde over (σ)}(ρ,z), and allelements of the vector Δ{tilde over ({overscore (σ)})} are equal. Asdescribed above, borehole corrections and a calibration can be appliedto each measurement, but here they are omitted for simplicity.

[0089] The vectors {overscore (v)}_(k) and Δ{tilde over ({overscore(σ)})} are related as follows:

{overscore (v)}_(k) =I[{overscore (S)}Δ{tilde over ({overscore (σ)})}]

[0090] in which {overscore (S)} is a diagonal matrix with each diagonalelement being the sensitivity function centered on the depth z_(k), forthe corresponding element of {overscore (v)}_(k), and the integraloperator I is defined by:I[F] = ∫_(−∞)^(+∞)  z∫₀^(+∞)  ρ  F(ρ, z).

[0091] Using the notationI_(mn)[F] = ∫_(z_(m − 1))^(z_(m))  z∫_(ρ_(n − 1))^(ρ_(n))  ρ  F(ρ, z)

[0092] to denote integrals of a function over the indicated limits ofintegration, it is apparent that${\overset{\_}{v}}_{k} = {\sum\limits_{m = {- M}}^{+ M}\quad {\sum\limits_{n = 1}^{N^{\prime}}\quad {I_{mn}\left\lbrack {\overset{\_}{\overset{\_}{S}}\Delta \overset{\_}{\overset{\sim}{\sigma}}} \right\rbrack}}}$

[0093] if ρ₀=0, ρ_(N′)=+∞, z_(−M−1)=−∞, and z_(M)=+∞. The equationdirectly above is an integral equation from which an estimate of Δ{tildeover (σ)}(ρ, z) can be calculated. With the definitionsρ_(n)*=(ρ_(n)+ρ_(n−1))/2 and Z_(m)*=(z_(m)+z_(m−1))/2 and making theapproximation Δ{tilde over (σ)}(ρ,z)=Δ{tilde over (σ)}(ρ_(n)*,z_(m)*)within the volumes associated with each value for m and n, it followsthat${\overset{\_}{v}}_{k} = {\sum\limits_{m = {- M}}^{+ M}\quad {\sum\limits_{n = 1}^{N^{\prime}}\quad {{I_{mn}\left\lbrack \overset{\_}{\overset{\_}{S}} \right\rbrack}\Delta {\overset{\_}{\hat{\sigma}}\left( {\rho_{n}^{*},z_{m}^{*}} \right)}}}}$

[0094] where N′≦N to ensure this system of equations is notunderdetermined. The unknown values Δ{circumflex over ({overscore(σ)})}(ρ_(n)*, z_(m)*) can then be determined by solving the above setof linear equations. It is apparent that the embodiment described in thesection entitled “REALIZATION OF THE TRANSFORMATION” is a special caseof the above for which M=0,N=N′=1.

[0095] Although the approximation Δ{tilde over (σ)}(ρ,z)=Δ{circumflexover (σ)}(ρ_(n)*, z_(m)*) (which merely states that Δ{tilde over(σ)}(ρ,z) is a piecewise constant function of ρ, z) is used in theimmediately above embodiment, such an approximation is not necessary.More generally, it is possible to expand Δ{tilde over (σ)}(ρ, z) using aset of basis functions, and to then solve the ensuing set of equationsfor the coefficients of the expansion. Specifically, suppose${{\Delta {\overset{\sim}{\sigma}\left( {\rho,z} \right)}} = {\sum\limits_{m = {- \infty}}^{\infty}\quad {\sum\limits_{n = {- \infty}}^{\infty}\quad {a_{mn}{\varphi_{mn}\left( {\rho,z} \right)}\quad {then}}}}},{{\overset{\_}{v}}_{k} = {\sum\limits_{m = {- \infty}}^{\infty}\quad {\sum\limits_{n = {- \infty}}^{\infty}\quad {{I\left\lbrack {\overset{\_}{\overset{\_}{S}}\varphi_{mn}} \right\rbrack}{\overset{\_}{a}}_{mn}}}}}$

[0096] where {overscore (a)}_(mn)=a_(mn)[1,1, . . . , 1]^(T). Somedesirable properties for the basis functions φ_(mn) are: 1) theintegrals I[{overscore (S)}φ_(mn)] in the above equation all exist; and,2) the system of equations for the coefficients a_(mn) is not singular.It is helpful to select the basis functions so that a minimal number ofterms is needed to form an accurate approximation to Δ{tilde over(σ)}(ρ, z).

[0097] The above embodiment is a special case for which the basisfunctions are unit step functions. In fact, employing the expansion${\Delta {\overset{\sim}{\sigma}\left( {\rho,z} \right)}} = {\sum\limits_{m = {- M}}^{+ M}{\sum\limits_{n = 1}^{N^{\prime}}{\Delta \quad {{{\overset{\_}{\hat{\sigma}}\left( {\rho_{n}^{*},z_{m}^{*}} \right)}\left\lbrack {{u\left( {z - z_{m}} \right)} - {u\left( {z - z_{m - 1}} \right)}} \right\rbrack}\left\lbrack {{u\left( {\rho - \rho_{n}} \right)} - {u\left( {\rho - \rho_{n - 1}} \right)}} \right\rbrack}}}}$

[0098] where u(•) denotes the unit step function leads directly to thesame system of equations${\overset{\_}{v}}_{k} = {\sum\limits_{m = {- M}}^{+ M}{\sum\limits_{n = 1}^{N^{\prime}}{{I_{m\quad n}\left\lbrack \overset{\overset{\_}{\_}}{S} \right\rbrack}\Delta \quad {\overset{\_}{\hat{\sigma}}\left( {\rho_{n}^{*},z_{m}^{*}} \right)}}}}$

[0099] given in the above embodiment. Specific values for M, N′, z_(m),and ρ_(n) needed to realize this embodiment of the invention depend onthe excitation frequency(ies), on the transmitter-receiver spacings thatare under consideration, and generally on the background conductivityand dielectric constant. Different values for z_(m) and ρ_(n) aregenerally used for different depth intervals within the same wellbecause the background medium parameters vary as a function of depth inthe well.

[0100] Solving the immediately above system of equations results inestimates of the average conductivity and dielectric constant within thevolume of the earth formation 215 corresponding to each integralI_(mn)[{overscore (S)}]. In an embodiment, the Least Mean Square methodis used to determine values for Δ{circumflex over ({overscore(σ)})}(ρ_(n)*z_(m)*) by solving the above system of equations. Manytexts on linear algebra list other techniques that may also be used.

[0101] Unlike other procedures previously used for processing MWD/LWDdata, the techniques of a disclosed embodiment account for dielectriceffects and provide for radial inhomogeneities in addition to beddinginterfaces by consistently treating the signal as a complex-valuedfunction of the conductivity and the dielectric constant. This procedureproduces estimates of one variable (i.e., the conductivity) are notcorrupted by effects of the other (i.e., the dielectric constant). Asmentioned in the above “SUMMARY OF THE INVENTION,” this result wasdeemed impracticable as a consequence of the “old assumptions.”

[0102] A series of steps, similar to those of FIG. 9, can be employed inorder to implement the embodiment for Multiple Sensors at MultipleDepths. Since the lookup table for I_(mn)[{overscore (S)}] needed torealize such an embodiment could be extremely large, these values areevaluated as needed in this embodiment. This can be done in a manneranalogous to the means described in the above section “REALIZATION OFTHE TRANSFORMATION” using the following formulae:${I_{m\quad n}\lbrack S\rbrack} = {\left. {\frac{1}{w_{0}}\frac{\partial w}{\partial{\overset{\sim}{\sigma}}_{m\quad n}}} \middle| {}_{\overset{\sim}{\alpha} = {\overset{\sim}{\alpha}}_{0}}\frac{\partial w}{\partial{\overset{\sim}{\sigma}}_{m\quad n}} \right|_{\overset{\sim}{\alpha} = {\overset{\sim}{\alpha}}_{0}} = {\lim\limits_{{\Delta {\overset{\sim}{\sigma}}_{m\quad n}}\rightarrow 0}{\frac{{w\left( {{\overset{\sim}{\sigma}}_{0} + {\Delta {\overset{\sim}{\sigma}}_{m\quad n}}} \right)} - {w\left( {\overset{\sim}{\sigma}}_{0} \right)}}{\left( {{\overset{\sim}{\sigma}}_{0} + {\Delta {\overset{\sim}{\sigma}}_{m\quad n}}} \right) - {\overset{\sim}{\sigma}}_{0}}.}}}$

[0103] where {tilde over (σ)}_(mn)=σ_(mn)+iωε_(mn) represents theconductivity and dielectric constant of the region of space over whichthe integral I_(mn)[S] is evaluated. In words, I_(mn)[S] can becalculated by evaluating the derivative of the measurement with respectto the medium parameters within the volume covered by the integration.Alternatively, one could evaluate I_(mn)[S] by directly carrying out theintegration as needed. This eliminates the need to store the values in alookup table.

[0104] While the above exemplary systems are described in the context ofan MWD/LWD system, it shall be understood that a system according to thedescribed techniques can be implemented in a variety of other loggingsystems such as wireline induction or wireline dielectric measurementsystems. Further in accordance with the disclosed techniques, it shouldbe understood that phase shift and attenuation can be combined in avariety of ways to produce a component sensitive to resistivity andrelatively insensitive to dielectric constant and a component sensitiveto dielectric constant and relatively insensitive to resistivity. In theinstance of MWD/LWD resistivity measurement systems, resistivity is thevariable of primary interest; as a result, phase shift and attenuationmeasurements can be combined to produce a component sensitive toresistivity and relatively insensitive dielectric constant.

[0105] Single Measurements at a Single Depth

[0106] One useful embodiment is to correlate (or alternatively equate) asingle measured value w₁ to a model that predicts the value of themeasurement as a function of the conductivity and dielectric constantwithin a prescribed region of the earth formation. The value for thedielectric constant and conductivity that provides an acceptablecorrelation (or alternatively solves the equation) is then used as thefinal result (i.e., correlated to the parameters of the earthformation). This procedure can be performed mathematically, orgraphically. Plotting a point on a chart such as FIG. 8 and thendetermining which dielectric value and conductivity correspond to it isan example of performing the procedure graphically. It can be concludedfrom the preceding sections, that Ŝ is the sensitivity of such anestimate of the dielectric constant and conductivity to perturbations ineither variable. Thus such a procedure results in an estimate for theconductivity that has no net sensitivity to the changes in thedielectric constant and an estimate for the dielectric constant that hasno net sensitivity to changes in the conductivity within the volume inquestion. This is a very desirable property for the results to have. Theutility of employing a single measurement at a single depth derives fromthe fact that data processing algorithms using minimal data as inputstend to provide results quickly and reliably. This procedure is a novelmeans of determining of one parameter (either the conductivity or thedielectric constant) with no net sensitivity to the other parameter.Under the old assumptions, this procedure would appear to not be usefulfor determining independent parameter estimates.

[0107] Iterative Forward Modeling and Dipping Beds

[0108] The analysis presented above has been carried out assuming a2-dimensional geometry where the volume P 225 in FIG. 2 is a solid ofrevolution about the axis of the tool. In MWD/LWD and wirelineoperations, there are many applications where such a 2-dimensionalgeometry is inappropriate. For example, the axis of the tool oftenintersects boundaries between different geological strata at an obliqueangle. Practitioners refer to the angle between the tool axis and avector normal to the strata as the relative dip angle. When the relativedip angle is not zero, the problem is no longer 2-dimensional. However,the conclusion that: 1) the attenuation measurement is sensitive to theconductivity in the same volume as the phase measurement is sensitive tothe dielectric constant; and, 2) an attenuation measurement is sensitiveto the dielectric constant in the same volume that the phase measurementis sensitive to the conductivity remains true in the more complicatedgeometry. Mathematically, this conclusion follows from theCauchy-Reimann equations which still apply in the more complicatedgeometry (see the section entitled “SENSITIVITY FUNCTIONS”). Thephysical basis for this conclusion is that the conduction currents arein quadrature (90 degrees out of phase) with the displacement currents.At any point in the formation, the conduction currents are proportionalto the conductivity and the displacement currents are proportional tothe dielectric constant.

[0109] A common technique for interpreting MWD/LWD and wireline data inenvironments with complicated geometry such as dipping beds is to employa model which computes estimates for the measurements as a function ofthe parameters of a hypothetical earth formation. Once model inputparameters have been selected that result in a reasonable correlationbetween the measured data and the model data over a given depthinterval, the model input parameters are then correlated to the actualformation parameters. This process is often referred to as “iterativeforward modeling” or as “Curve Matching,” and applying it in conjunctionwith the old assumptions, leads to errors because the volumes in whicheach measurement senses each variable have to be known in order toadjust the model parameters appropriately.

[0110] The algorithms discussed in the previous sections can also beadapted for application to data acquired at non-zero relative dipangles. Selecting the background medium to be a sequence of layershaving the appropriate relative dip angle is one method for so doing.

[0111] Transformations For a Resistivity-Dependent Dielectric Constant

[0112] In the embodiments described above, both the dielectric constantand conductivity are treated as independent quantities and the intent isto estimate one parameter with minimal sensitivity to the other. Asshown in FIG. 1, there is empirical evidence that the dielectricconstant and the conductivity can be correlated. Such empiricalrelationships are widely used in MWD/LWD applications, and when theyhold, one parameter can be estimated if the other parameter is known.

[0113] This patent application shows that: 1) an attenuation measurementis sensitive to the conductivity in the same volume of an earthformation as the phase measurement is sensitive to the dielectricconstant; and, 2) the attenuation measurement is sensitive to thedielectric constant in the same volume that the phase measurement issensitive to the conductivity. A consequence of these relationships isthat it is not generally possible to derive independent estimates of theconductivity from a phase and an attenuation measurement even if thedielectric constant is assumed to vary in a prescribed manner as afunction of the conductivity. The phrase “not generally possible” isused above because independent estimates from each measurement can bestill be made if the dielectric constant doesn't depend on theconductivity or if the conductivity and dielectric constant of earthformation are practically the same at all points within the sensitivevolumes of both measurements. Such conditions represent special caseswhich are not representative of conditions typically observed withinearth formations.

[0114] Even though two independent estimates of the conductivity are notgenerally possible from a single phase and a single attenuationmeasurement, it is still possible to derive two estimates of theconductivity from a phase and an attenuation measurement given atransformation to convert the dielectric constant into a variable thatdepends on the resistivity. For simplicity, consider a device such asthat of FIG. 3. Let the complex number w₁ denote an actual measurement(i.e., the ratio of the voltage at receiver 307 relative the voltage atreceiver 309, both voltages induced by current flowing throughtransmitter 305). Let the complex number w denote the value of saidmeasurement predicted by a model of the tool 320 in a prescribed earthformation 315. For further simplicity, suppose the model is as describedabove in the section “REALIZATION OF THE TRANSFORMATION.” Then,${w \equiv {w\left( {\sigma,{ɛ(\sigma)}} \right)}} = {\left( \frac{L_{1}}{L_{2}} \right)^{3}\frac{{\exp \left( {\quad k\quad L_{2}} \right)}\left( {1 - {\quad k\quad L_{2}}} \right)}{{\exp \left( {\quad k\quad L_{1}} \right)}\left( {1 - {\quad k\quad L_{1}}} \right)}}$

[0115] where the wave number k≡k(σ,ε(σ))={square root}{square root over(iωμ(σ+iωε(σ)))}, and the dependence of the dielectric constant ε on theconductivity σ is accounted for by the function ε(σ). Differentfunctions ε(σ) can be selected for different types of rock. Let σ_(P)and σ_(A) denote two estimates of the conductivity based on a phase andan attenuation measurement and a model such as the above model. Theestimates can be determined by solving the system of equations

0=|w ₁ |−|w(σ_(A),ε(σ_(P)))|

0=phase(w ₁)−phase(w(σ_(P),ε(σ_(A)))).

[0116] The first equation involves the magnitude (a.k.a. theattenuation) of the measurement and the second equation involves thephase (a.k.a. the phase shift) of the measurement. Note that thedielectric constant of one equation is evaluated using the conductivityof the other equation.

[0117] This disclosed technique does not make use of the “oldassumptions.” Instead, the attenuation conductivity is evaluated using adielectric value consistent with the phase conductivity and the phaseconductivity is evaluated using a dielectric constant consistent withthe attenuation conductivity. These conductivity estimates are notindependent because the equations immediately above are coupled (i.e.,both variables appear in both equations). The above described techniquesrepresent a substantial improvement in estimating two resistivity valuesfrom a phase and an attenuation measurement given a priori informationabout the dependence of the dielectric constant on the conductivity. Itcan be shown that the sensitivity functions for the conductivityestimates σ_(A)and σ_(P) are S′ and S″, respectively if the perturbationto the volume P 225 is consistent with the assumed dependence of thedielectric constant on the conductivity.

[0118] It will be evident to those skilled in the art that a morecomplicated model can be used in place of the simplifying assumptions.Such a model may include finite antennas, metal or insulating mandrels,formation inhomogeneities and the like. In addition, other systems ofequations could be defined such as ones involving the real and imaginaryparts of the measurements and model values. As in previous sections ofthis disclosure, calibration factors and borehole corrections may beapplied to the raw data.

[0119] The foregoing disclosure and description of the variousembodiments are illustrative and explanatory thereof, and variouschanges in the descriptions and attributes of the system, theorganization of the measurements, transmitter and receiverconfigurations, and the order and timing of steps taken, as well as inthe details of the illustrated system may be made without departing fromthe spirit of the invention.

What we claim is:
 1. A method of determining electrical parameters of anearth formation through which a borehole is drilled, the methodcomprising the steps of: exciting and measuring a plurality ofelectrical signals that penetrate an earth formation using one or moretransmitting antennas and one or more receiving antennas placed within aborehole within the earth formation; estimating a plurality ofbackground values, each background value corresponding to an electricalparameter of the plurality of electrical parameters and based upon atleast one corresponding electrical signal of the plurality of electricalsignals and on a model of the earth formation; and, calculating theplurality of electrical parameters by correlating the plurality ofelectrical parameters to the corresponding background values such thatresulting estimates of the plurality of electrical parameters areindependent of each other.
 2. The method of claim 1, wherein thecalculating step comprises the steps of: computing a plurality ofmeasurement sensitivity values corresponding to any electrical signal ofthe plurality of electrical signals based upon a change in one or moreof the electrical parameters of the earth formation from one or more ofthe background values within a volume of the earth formation;determining a plurality of perturbation values corresponding to theelectrical parameters of the earth formation from one or more of thebackground values using some of the plurality of electrical signals andsome of the plurality of measurement sensitivity values; and correlatingan electrical parameter to the sum of a corresponding perturbation valueand a corresponding background value.
 3. The method of claim 2, whereinthe plurality of measurement sensitivity values are stored in a lookuptable as a function of the background medium parameters and are accessedfrom the lookup table.
 4. The method of claim 2, wherein the pluralityof measurement sensitivity values are calculated based on the positionof the transmitting and receiving antennas along the borehole and on thebackground medium parameters on command as needed.
 5. The method ofclaim 2, wherein the perturbation values are determined using a formula:${{\Delta \quad \hat{\sigma}} = {\frac{I\left\lbrack {S\quad \Delta \overset{\sim}{\sigma}} \right\rbrack}{I\lbrack S\rbrack} = {{I\left\lbrack {\hat{S}\quad \Delta \overset{\sim}{\sigma}} \right\rbrack} = {\frac{1}{I\lbrack S\rbrack}\left( {\frac{{cw}_{1} - w_{bh}}{w_{0}} - 1} \right)}}}};$

where Δ{circumflex over (σ)} is a complex number including a real partand an imaginary part, the real part of which is correlated to thedifference between a conductivity of the earth formation and aconductivity of a background medium, and the imaginary part of which iscorrelated to the product of the excitation frequency and the differencebetween a dielectric constant of the earth formation and a dielectricconstant of the background medium; the integral operator I[•] representsan integral over all space of an argument for the integral operatorI[•]; the function S represents a complex-valued sensitivity functionfor attenuation and phase shift measurements to variations in electricalparameters of the earth formation as a function of position within theearth formation; a symbol Δ{tilde over (σ)} is a complex-valued functionof position within the earth formation, the real part of Δ{tilde over(σ)} is the difference between the electrical conductivity of the earthformation at a same location and the electrical conductivity of thebackground medium at the same location, and the imaginary part ofΔ{tilde over (σ)} is proportional to the difference between thedielectric constant of the earth formation at said location and thedielectric constant of the background medium at said location; thecalibration factor, c, is a complex number that accounts forirregularities of the exciting and measuring step; w₁ is a complexnumber representing a measurement of a plurality of electrical signals;w₀ represents an expected value of w₁ in the background medium; andw_(bh) represents borehole effects.
 6. The method of claim 1, furthercomprising the step of: applying a plurality of calibration factors tothe plurality of electrical signals to account for irregularities of theexciting and measuring step.
 7. The method of claim 1, wherein theelectrical parameter(s) comprise a resistivity value and a dielectricconstant.
 8. The method of claim 1, where the electrical parameter(s) tobe determined comprise a resistivity value, independent of a dielectricconstant.
 9. The method of claim I, where the electrical parameter(s) tobe determined comprise a dielectric constant, independent of aresistivity value.
 10. The method of claim 1, wherein the electricalsignals are excited at a frequency between 5 kHz and 2 GHz.
 11. Themethod of claim 1, wherein the plurality of background values arecomputed using data at multiple excitation frequencies.
 12. The methodof claim 1, wherein the electrical signal(s) comprise an attenuationmeasurement and a phase shift measurement.
 13. The method of claim 1,wherein the electrical signal(s) are derived from a ratio of thevoltages at two or more receiver antennas.
 14. The method of claim 1,wherein the transmitting antennas consist of a plurality of coilsconnected in electrical series, the receiving antennas consist of aplurality of coils connected in electrical series, and the electricalsignal(s) are the real and imaginary parts of the voltage across thereceiver antennas relative to a current at the transmitter antennas. 15.The method of claim 1, wherein the electrical signal(s) are measured atmultiple points along the borehole.
 16. The method of claim 1, theestimating background values step comprising the steps of: averagingeach measured electrical signal over a depth interval appropriate to theresolution characteristics of the measured electrical signal; andcorrelating the background values to parameters of the model of theearth formation such that the corresponding average of the electricalsignals from the model are comparable to each average electrical signalvalue computed in the averaging step.
 17. The method of claim 1, whereinthe model for the earth formation used is a model of a homogeneousmedium.
 18. The method of claim 1, wherein the model for the earthformation is a model of a medium with inhomogeneities.
 19. The method ofclaim 1, further comprising the step of: adjusting an estimate of theplurality of background values resulting from the estimating step forerrors caused by the presence of the borehole.
 20. The method of claim1, further comprising the step of: adjusting the plurality of electricalsignals for errors caused by the borehole.
 21. The method of claim 1,the estimating background values step comprising the steps of:independently calculating estimates of the plurality of backgroundvalues individually from each of the plurality of electrical signals ata given point along the borehole; and, averaging said independentestimates from said independently calculating step over an intervalalong the borehole.
 22. A method of calculating electrical parametervalues of an earth formation penetrated by a borehole, the methodcomprising the steps of: transforming a measured phasor signal of aplurality of measured phasor signals so that a first component of atransformed phasor signal is relatively sensitive to a first electricalparameter and relatively insensitive to a second electrical parameterwithin each volume of a plurality of volumes within an earth formation;and, correlating the first component to the first electrical parameter.23. The method of claim 22, further comprising the steps of:transforming the measured phasor signal so that a second component ofthe transformed phasor signal is relatively sensitive to the secondelectrical parameter and relatively insensitive to the first electricalparameter within each volume of the plurality of volumes within theearth formation; and correlating the second component of the secondelectrical parameter.
 24. The method of claim 22, wherein the firstelectrical parameter comprises a resistivity and the second electricalparameter comprises a dielectric constant.
 25. The method of claim 22,wherein the measured phasor signal comprises a real part and animaginary part.
 26. The method of claim 22, wherein the measured phasorsignal comprises an attenuation value and a phase shift value.
 27. Themethod of claim 22, the transforming step comprising the step of:correlating one or both of the first electrical parameter and the secondelectrical parameter to a model which predicts the measured phasorsignal in a homogeneous medium.
 28. A method of determining a firstelectrical parameter of an earth formation through which a borehole isdrilled, the method comprising the steps of: transforming a secondelectrical parameter of the earth formation into a variable that dependson the first electrical parameter; and, estimating two values for thefirst electrical parameter from the properties of a measured electricalsignal in a manner consistent with each property of the measuredelectrical signal sensing the first electrical parameter and the secondelectrical parameter in different volumes and also consistent with thetransforming step.
 29. The method of claim 28, wherein the measuredelectrical signal comprises an attenuation measurement and a phase shiftmeasurement between a first receiver coil and a second receiver coil.30. The method of claim 28, wherein the first electrical parametercomprises a resistivity of the earth formation and the second electricalparameter comprises a dielectric constant of the earth formation. 31.The method of claim 30, wherein the estimates for the two resistivityvalues are determined by simultaneously solving a first equation,0=|w₁|−|w(σ_(A), ε(σ_(P)))|, and a second equation,0=phase(w₁)−phase(w(σ_(P),ε(σ_(A)))), where σ_(A) and σ_(P) representthe reciprocals of the resistivity values; the first equation involves amagnitude of the measured electrical signal at a given frequency ofexcitation and the second equation involves a phase of the measuredsignal; a function ε(•) represents a correlation between the dielectricconstant and the resistivity and is evaluated in σ_(P) in the firstequation and in σ_(A) in the second equation; w₁ represents an actualmeasurement in the form of a complex number, and a function w(σ,ε)represents a model which estimates w₁.
 32. The method of claim 28,wherein the first electrical parameter comprises a dielectric constantof the earth formation and the second electrical parameter comprises aresistivity of the earth formation.
 33. The method of claim 28, whereinthe transforming and estimating steps are repeated at a plurality ofpoints along the borehole.
 34. A computing system for calculatingelectrical parameter values of an earth formation penetrated by aborehole, comprising: a first means for transforming a measured phasorsignal of a plurality of measured phasor signals so that a firstcomponent of a transformed phasor signal is relatively sensitive to afirst electrical parameter and relatively insensitive to a secondelectrical parameter within each volume of a plurality of volumes withinan earth formation; and, a first means for correlating the firstcomponent to the first electrical parameter.
 35. The computing system ofclaim 34, further comprising: a second means for transforming themeasured phasor signal so that a second component of the transformedphasor signal is relatively sensitive to the second electrical parameterand relatively insensitive to the first electrical parameter within eachvolume of the plurality of volumes within the earth formation; and asecond means for correlating the second component to the secondelectrical parameter.
 36. The computing system of claim 34, wherein thefirst electrical parameter comprises a resistivity of the earthformation and the second electrical parameter comprises a dielectricconstant of the earth formation.
 37. The computing system of claim 34,wherein the measured phasor signal comprises a real part and animaginary part.
 38. The computing system of claim 34, wherein themeasured phasor signal comprises an attenuation value and a phase shiftvalue.
 39. The computing system of claim 34, the first or second meanscomprising: a means for correlating one or both of the first electricalparameter and the second electrical parameter to a model which predictsthe measured phasor signal in a homogeneous medium.
 40. A computingsystem for determining a first electrical parameter of an earthformation through which a borehole is drilled, comprising: a means fortransforming a second electrical parameter of the earth formation into avariable that depends on the first electrical parameter; and, a meansfor estimating two values for the first electrical parameter from theproperties of a measured electrical signal in a manner consistent witheach property of the measured electrical signal sensing the firstelectrical parameter and the second electrical parameter in differentvolumes and also consistent with the transforming step.
 41. Thecomputing system of claim 40, wherein the measured electrical signalcomprises an attenuation measurement and a phase shift measurementbetween a first receiver coil and a second receiver coil.
 42. Thecomputing system of claim 40, wherein the first electrical parametercomprises a resistivity of the earth formation and the second electricalparameter comprises a dielectric constant of the earth formation. 43.The computing system of claim 42, wherein the estimates for the tworesistivity values are determined by simultaneously solving a firstequation, 0=|w₁|−|w(σ_(A),ε(σ_(P)))|, and a second equation,0=phase(w₁)−phase(w(σ_(P),ε(σ_(A)))), where σ_(A) and σ_(P) representthe reciprocals of the resistivity values; the first equation involves amagnitude of the measured electrical signal at a given frequency ofexcitation and the second equation involves a phase of the measuredsignal; a function ε(•) represents a correlation between the dielectricconstant and the resistivity and is evaluated in σ_(P) in the firstequation and in σ_(A) in the second equation; w₁ represents an actualmeasurement in the form of a complex number, and a function w(σ, ε)represents a model which estimates w₁.
 44. The computing system of claim40, wherein the first electrical parameter comprises a dielectricconstant of the earth formation and the second electrical parametercomprises a resistivity of the earth formation.
 45. The computing systemof claim 40, wherein the transforming means and the estimating means areemployed at a plurality of points along the borehole.